The Principal Component Analysis Network (PCANet), which is one of the recently proposed deep learning architectures, achieves the state-of-the-art classification accuracy in various databases. However, the performance of PCANet may be degraded when dealing with color images. In this paper, a Quaternion Principal Component Analysis Network (QPCANet), which is an extension of PCANet, is proposed for color images classification. Compared to PCANet, the proposed QPCANet takes into account the spatial distribution information of color images and ensures larger amount of intra-class invariance of color images. Experiments conducted on different color image datasets such as Caltech-101, UC Merced Land Use, Georgia Tech face and CURet have revealed that the proposed QPCANet achieves higher classification accuracy than PCANet.
Recently, quaternion collaborative representation-based classification (QCRC) and quaternion sparse representation-based classification (QSRC) have been proposed for color face recognition. They can obtain correlation information among different color channels. However, their performance is unstable in different conditions. For example, QSRC performs better than than QCRC on some situations but worse on other situations. To benefit from quaternion-based $e_2$-norm minimization in QCRC and quaternion-based $e_1$-norm minimization in QSRC, we propose the quaternion-based adaptive representation …
Fujian Normal University（福建师范大学）；Center for Applied Mathematics of Fujian province (FJNU)（福建省应用数学中心（福建师范大学））；Jiangsu Normal University（江苏师范大学）；The Research Institute of Mathematical Science, Jiangsu Normal University（江苏师范大学数学科学研究院）；Fuzhou University of International Studies and Trade（福州外语外贸学院）
To address the non-negativity dropout problem of quaternion models, a novel quasi non-negative quaternion matrix factorization (QNQMF) model is presented for color image processing. To implement QNQMF, the quaternion projected gradient algorithm and the quaternion alternating direction method of multipliers are proposed via formulating QNQMF as the non-convex constraint quaternion optimization problems. Some properties of the proposed algorithms are studied. The numerical experiments on the color image reconstruction show that these algorithms encoded on the quaternion perform better than thes…
China University of Mining and Technology（中国矿业大学）；Jiangsu Normal University（江苏师范大学）
A sample-relaxed two-dimensional color principal component analysis (SR-2DCPCA) approach is presented for face recognition and image reconstruction based on quaternion models. A relaxation vector is automatically generated according to the variances of training color face images with the same label. A sample-relaxed, low-dimensional covariance matrix is constructed based on all the training samples relaxed by a relaxation vector, and its eigenvectors corresponding to the $r$ largest eigenvalues are defined as the optimal projection. The SR-2DCPCA aims to enlarge the global variance rather than…
University of Texas at San Antonio（德克萨斯大学圣安东尼奥分校）；City University of New York / CSI（纽约城市大学 / 斯塔滕岛学院）
Color in an image is resolved into 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discre…
Blind Deconvolution for Color Images Using Normalized Quaternion Kernels
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University（数学科学学院、智能计算与应用重点实验室、同济大学）；Department of Mathematics, Hong Kong Baptist University（数学系、香港 Baptist 大学）；School of Mathematics and Statistics & RIIS, Jiangsu Normal University（数学与统计学学院及RIIS、江苏师范大学）
Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction
School of Mathematical Science, Inner Mongolia University（内蒙古大学数学科学学院）；The Chinese University of Hong Kong（香港中文大学）；Hong Kong Baptist University（香港浸会大学）
University of Macau（澳门大学）；Beijing Normal University at Zhuhai（北京师范大学珠海校区）
In this article, a robust color-edge feature extraction method based on the Quaternion Hardy filter is proposed. The Quaternion Hardy filter is an emerging edge detection theory. It is along with the Poisson and conjugate Poisson smoothing kernels to handle various types of noise. Combining with the Quaternion Hardy filter, Jin's color gradient operator and Hough transform, the color-edge feature detection algorithm is proposed and applied to the lane marking detection. Experiments are presented to demonstrate the validity of the proposed algorithm. The results are accurate and robust with res…
National Institute of Standards and Technology（美国国家标准与技术研究院）
In this paper, we investigate the use of 3D surface geometry for face recognition and compare it to one based on color map information. The 3D surface and color map data are from the CAESAR anthropometric database. We find that the recognition performance is not very different between 3D surface and color map information using a principal component analysis algorithm. We also discuss the different techniques for the combination of the 3D surface and color map information for multi-modal recognition by using different fusion approaches and show that there is significant improvement in results. …
University of Macau（澳门大学）；Beijing Normal University（北京师范大学）；Shandong University of Finance and Economics（山东财经大学）
This paper presents a robust filter called quaternion Hardy filter (QHF) for color image edge detection. The QHF can be capable of color edge feature enhancement and noise resistance. It is flexible to use QHF by selecting suitable parameters to handle different levels of noise. In particular, the quaternion analytic signal, which is an effective tool in color image processing, can also be produced by quaternion Hardy filtering with specific parameters. Based on the QHF and the improved Di Zenzo gradient operator, a novel color edge detection algorithm is proposed. Importantly, it can be effic…
University of Texas at San Antonio（德克萨斯大学圣安东尼奥分校）；City University of New York / CSI（纽约市立大学/斯塔滕岛学院）
In this paper a quaternion approach of enhancement method is proposed in which color in the image is considered as a single entity. This new method is referred as the alpha-rooting method of color image enhancement by the two-dimensional quaternion discrete Fourier transform (2-D QDFT) followed by a spatial transformation. The results of the proposed color image enhancement method are compared with its counterpart channel-by-channel enhancement algorithm by the 2-D DFT. The image enhancements are quantified to the enhancement measure that is based on visual perception referred as the color enh…
Depth-Aided Color Image Inpainting in Quaternion Domain
Graduate School of Engineering Science, Osaka University（大阪大学工程科学研究生院）；Institute of Engineering, Tokyo University of Agriculture and Technology（东京农工大学工程学院）
Scene Background Initialization (SBI) is one of the challenging problems in computer vision. Dynamic mode decomposition (DMD) is a recently proposed method to robustly decompose a video sequence into the background model and the corresponding foreground part. However, this method needs to convert the color image into the grayscale image for processing, which leads to the neglect of the coupling information between the three channels of the color image. In this study, we propose a quaternion-based DMD (Q-DMD), which extends the DMD by quaternion matrix analysis, so as to completely preserve the…
For many years, channels of a color image have been processed individually, or the image has been converted to grayscale one with respect to color image processing. Pure quaternion representation of color images solves this issue as it allows images to be processed in a holistic space. Nevertheless, it brings additional costs due to the extra fourth dimension. In this paper, we propose an approach for representing color images with full quaternion numbers that enables us to process color images holistically without additional cost in time, space and computation. With taking auto- and cross-cor…
Neural networks in the real domain have been studied for a long time and achieved promising results in many vision tasks for recent years. However, the extensions of the neural network models in other number fields and their potential applications are not fully-investigated yet. Focusing on color images, which can be naturally represented as quaternion matrices, we propose a quaternion convolutional neural network (QCNN) model to obtain more representative features. In particular, we redesign the basic modules like convolution layer and fully-connected layer in the quaternion domain, which can…
Higher-order singular value decomposition (HOSVD) is one of the most efficient tensor decomposition techniques. It has the salient ability to represent high_dimensional data and extract features. In more recent years, the quaternion has proven to be a very suitable tool for color pixel representation as it can well preserve cross-channel correlation of color channels. Motivated by the advantages of the HOSVD and the quaternion tool, in this paper, we generalize the HOSVD to the quaternion domain and define quaternion-based HOSVD (QHOSVD). Due to the non-commutability of quaternion multiplicati…
The University of Texas at San Antonio（德克萨斯大学圣安东尼奥分校）；City University of New York / CSI（纽约城市大学/斯塔滕岛学院）
Over the past century, a correlation has been an essential mathematical technique utilized in engineering sciences, including practically every signal/image processing field. This paper describes an effective method of calculating the correlation function of signals and color images in quaternion algebra. We propose using the quaternions with a commutative multiplication operation and defining the corresponding correlation function in this arithmetic. The correlation between quaternion signals and images can be calculated by multiplying two quaternion DFTs of signals and images. The complexity…
University of Macau（澳门大学）；Zhejiang Normal University（浙江师范大学）
This paper addresses the color image completion problem in accordance with low-rank quatenrion matrix optimization that is characterized by sparse regularization in a transformed domain. This research was inspired by an appreciation of the fact that different signal types, including audio formats and images, possess structures that are inherently sparse in respect of their respective bases. Since color images can be processed as a whole in the quaternion domain, we depicted the sparsity of the color image in the quaternion discrete cosine transform (QDCT) domain. In addition, the representatio…
In this paper, we study color image inpainting as a pure quaternion matrix completion problem. In the literature, the theoretical guarantee for quaternion matrix completion is not well-established. Our main aim is to propose a new minimization problem with an objective combining nuclear norm and a quadratic loss weighted among three channels. To fill the theoretical vacancy, we obtain the error bound in both clean and corrupted regimes, which relies on some new results of quaternion matrices. A general Gaussian noise is considered in robust completion where all observations are corrupted. Moti…
School of Mathematics and Statistics（数学与统计学学院）；Beijing Jiaotong University（北京交通大学）；College of Information and Management Science（信息与管理科学学院）；Henan Agricultural University（河南农业大学）
Jiangsu Normal University（江苏师范大学）；Inner Mongolia University（内蒙古大学）；The University of Hong Kong（香港大学）；University of Electronic Science and Technology of China（电子科技大学）
The image nonlocal self-similarity (NSS) prior refers to the fact that a local patch often has many nonlocal similar patches to it across the image and has been widely applied in many recently proposed machining learning algorithms for image processing. However, there is no theoretical analysis on its working principle in the literature. In this paper, we discover a potential causality between NSS and low-rank property of color images, which is also available to grey images. A new patch group based NSS prior scheme is proposed to learn explicit NSS models of natural color images. The numerical…
Laboratoire MIA, Bâtiment Pascal, Pôle Sciences et Technologies, Université de La Rochelle（拉罗谢尔大学MIA实验室）；Center for Ubiquitous Computing, Faculty of Information Technology and Electrical Engineering, University of Oulu（奥卢大学无处不在计算中心）；Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学）
Convolutional neural networks (CNN) have recently achieved state-of-the-art results in various applications. In the case of image recognition, an ideal model has to learn independently of the training data, both local dependencies between the three components (R,G,B) of a pixel, and the global relations describing edges or shapes, making it efficient with small or heterogeneous datasets. Quaternion-valued convolutional neural networks (QCNN) solved this problematic by introducing multidimensional algebra to CNN. This paper proposes to explore the fundamental reason of the success of QCNN over …
Image splicing detection is of fundamental importance in digital forensics and therefore has attracted increasing attention recently. In this paper, a color image splicing detection approach is proposed based on Markov transition probability of quaternion component separation in quaternion discrete cosine transform (QDCT) domain and quaternion wavelet transform (QWT) domain. Firstly, Markov features of the intra-block and inter-block between block QDCT coefficients are obtained from the real part and three imaginary parts of QDCT coefficients respectively. Then, additional Markov features are …
Escuela Superior Politécnica del Litoral（滨海理工学院）；Universidade Estadual de Campinas（坎皮纳斯州立大学）
The field of neural networks has seen significant advances in recent years with the development of deep and convolutional neural networks. Although many of the current works address real-valued models, recent studies reveal that neural networks with hypercomplex-valued parameters can better capture, generalize, and represent the complexity of multidimensional data. This paper explores the quaternion-valued convolutional neural network application for a pattern recognition task from medicine, namely, the diagnosis of acute lymphoblastic leukemia. Precisely, we compare the performance of real-va…
Nanyang Technological University (NTU)（南洋理工大学）；Advanced Digital Sciences Center (ADSC)（高级数字科学中心）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Sky/cloud imaging using ground-based Whole Sky Imagers (WSI) is a cost-effective means to understanding cloud cover and weather patterns. The accurate segmentation of clouds in these images is a challenging task, as clouds do not possess any clear structure. Several algorithms using different color models have been proposed in the literature. This paper presents a systematic approach for the selection of color spaces and components for optimal segmentation of sky/cloud images. Using mainly principal component analysis (PCA) and fuzzy clustering for evaluation, we identify the most suitable col…
Shanghai University（上海大学）；Research Institute of USV Engineering（无人艇工程研究院）；Jiangsu Normal University（江苏师范大学）；Research Institute of Mathematical Science（数学科学研究院）
Quaternion singular value decomposition (QSVD) is a robust technique of digital watermarking which can extract high quality watermarks from watermarked images with low distortion. In this paper, QSVD technique is further investigated and an efficient robust watermarking scheme is proposed. The improved algebraic structure-preserving method is proposed to handle the problem of "explosion of complexity" occurred in the conventional QSVD design. Secret information is transmitted blindly by incorporating in QSVD two new strategies, namely, coefficient pair selection and adaptive embedding. Unlike …
We study the problem of tensor robust principal component analysis (TRPCA), which aims to separate an underlying low-multilinear-rank tensor and a sparse outlier tensor from their sum. In this work, we propose a fast non-convex algorithm, coined Robust Tensor CUR (RTCUR), for large-scale TRPCA problems. RTCUR considers a framework of alternating projections and utilizes the recently developed tensor Fiber CUR decomposition to dramatically lower the computational complexity. The performance advantage of RTCUR is empirically verified against the state-of-the-arts on the synthetic datasets and is…
